I've up to a point dimly understood differentiation - the idea that the slope of a graph is its rate of change, and that you can differentiate an equation like x3 + 5x2 +12x -7 quite simply by following the rules - the x3 goes to 3x2, the 5x2 goes to 10x, the x goes to 12 and the 7 disappears. You can check this very easily by just cut and pasting my first formula into this brilliant derivative-calculator site Wonderful though this site is it doesn't actually explain why this is happening - it explains the individual rules, but not why the rule works.
It wasn't until I started reading Classical Mechanics: The Theoretical Minimum (Theoretical Minimum 1) by Leonard Susskind and George Hrabovsky, and then checked my answers to the differentiation question that I actually understood the marvelous mathematical sleight of hand that differentiation is.
I had to do it in my own hard way by looking at each line of the explanation, then matching up bits of equation with the next line to see what was different or missing. It was then that I really understood what that delta t, that little bit of infinitesimal time actually does. So that I never forget it I'm going to lay it out all in glorious colour here.
What we're actually doing with differentiation is taking a function like this one: f(x) = x3 + 5x2 +12x -7 and then writing out the equation again as f(x+𝛥t) - basically adding 𝛥t on every x as so:
f(x+𝛥t) = (x+𝛥t)3 + 5(x+𝛥t)2 +12(x+𝛥t) -7
which we then multiply out to make things easier to combine. (𝛥t or delta t as it is pronounced is just a very small bit of... time.) Of course I began to get into real grief here just sorting this out and had to visit another fantastic cheat site -Symbolab that did the expansion for me - and showed me how it got there. I had forgotten that there were formulae for multiplying out things like (x+𝛥t)3
f(x+𝛥t) = x3 +3x2𝛥t+3x𝛥t2+𝛥t3+ 5x2+10x𝛥t+5𝛥t2+12x+12𝛥t-7
So there we go with the expanded f(x+𝛥t). Then we write out f(x) - f(x+𝛥t) using these expanded formulae and take one away from the other.
f(x+𝛥t)- f(x) = x3 +3x2𝛥t+3x𝛥t2+𝛥t3+ 5x2+10x𝛥t+5𝛥t2-12x+12𝛥t+7- x3 - 5x2 +12x -7
The coloured pairs like x3 from both f(x) and f(x+𝛥t) nullify each other so you're left with:
f(x+𝛥t)- f(x) =3x2𝛥t+3x𝛥t2+𝛥t3+10x𝛥t+5𝛥t2+12𝛥t
Now the really clever bit - divide both sides by 𝛥t and you get 3x2+10x+12. All the multiples of delta t vanish as you go to zero - what you are doing is "taking the limit" as delta t heads for zero. I should write it out properly really but I need a better mathematical editor to do that but here it is bodged up with the space bar to line it up on two lines.
lim f(x+-𝛥t) -f(t) = 3x2+10x+12
𝛥t->0 𝛥t
Looking at the writing out of f(x) - f(x+𝛥t) using these expanded formulae above I realised I had got all the plus and minus signs in the wrong place, but the gist of it is if you take f(x) = x3 + 5x2 +12x -7 away from f(x+𝛥t) = (x+𝛥t)3 + 5(x+𝛥t)2 +12(x+𝛥t) -7 all you get left with is the bits that have 𝛥t in them. Well it makes sense to me now - but it's been days, days.
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